If log` ((5c)/(a)) , log ((3b)/(5c)) and log ((a)/(3b))` are n an A.P., where a, b and c are in a GP, then a, b and c, are the lengths of sides of

Options

A. An isosceles triangle
B. An equilateral triangle
C. A scalene triangle
D. None of these

Answer

To solve the problem step by step, we need to analyze the given information and apply the properties of logarithms and geometric progressions.

### Step 1: Understand the Given Information
We have three logarithmic expressions:
1. \( \log\left(\frac{5c}{a}\right) \)
2. \( \log\left(\frac{3b}{5c}\right) \)
3. \( \log\left(\frac{a}{3b}\right) \)

These three terms are in Arithmetic Progression (A.P.). Additionally, \( a, b, c \) are in Geometric Progression (G.P.).

### Step 2: Set Up the A.P. Condition
For three terms to be in A.P., the difference between the first and second terms must equal the difference between the second and third terms:
\[ \log\left(\frac{3b}{5c}\right) - \log\left(\frac{5c}{a}\right) = \log\left(\frac{a}{3b}\right) - \log\left(\frac{3b}{5c}\right) \]

### Step 3: Simplify Using Logarithmic Properties
Using the property of logarithms \( \log\left(\frac{m}{n}\right) = \log(m) - \log(n) \), we can rewrite the equation:
\[ \log(3b) - \log(5c) - \left(\log(5c) - \log(a)\right) = \log(a) - \log(3b) - \left(\log(3b) - \log(5c)\right) \]

This simplifies to:
\[ \log(3b) - 2\log(5c) + \log(a) = \log(a) - 2\log(3b) + \log(5c) \]

### Step 4: Rearranging the Equation
Rearranging the terms gives:
\[ \log(3b) + \log(5c) = 2\log(5c) + 2\log(3b) \]
This leads to:
\[ \log(3b) + \log(5c) = 2\log(3b) + 2\log(5c) \]

### Step 5: Further Simplification
This can be simplified further, leading us to:
\[ 3b = 5c \quad \text{(Cubing both sides)} \]
Thus, we have:
\[ (3b)^3 = (5c)^3 \implies 27b^3 = 125c^3 \implies \frac{b^3}{c^3} = \frac{125}{27} \implies \frac{b}{c} = \frac{5}{3} \]

### Step 6: Use the G.P. Condition
Since \( a, b, c \) are in G.P., we know:
\[ b^2 = ac \]
Substituting \( b = \frac{5}{3}c \) into the equation:
\[ \left(\frac{5}{3}c\right)^2 = a \cdot c \]
This simplifies to:
\[ \frac{25}{9}c^2 = ac \implies a = \frac{25}{9}c \]

### Step 7: Determine the Sides of the Triangle
Now we have:
- \( a = \frac{25}{9}c \)
- \( b = \frac{5}{3}c \)
- \( c = c \)

### Step 8: Check the Triangle Inequality
To check if these can form a triangle, we need to verify the triangle inequalities:
1. \( a + b > c \)
2. \( a + c > b \)
3. \( b + c > a \)

Calculating:
1. \( \frac{25}{9}c + \frac{5}{3}c > c \)
- \( \frac{25}{9}c + \frac{15}{9}c > c \)
- \( \frac{40}{9}c > c \) (True)

2. \( \frac{25}{9}c + c > \frac{5}{3}c \)
- \( \frac{25}{9}c + \frac{9}{9}c > \frac{15}{9}c \)
- \( \frac{34}{9}c > \frac{15}{9}c \) (True)

3. \( \frac{5}{3}c + c > \frac{25}{9}c \)
- \( \frac{15}{9}c + \frac{9}{9}c > \frac{25}{9}c \)
- \( \frac{24}{9}c > \frac{25}{9}c \) (False)

### Conclusion
Since the third inequality fails, \( a, b, c \) cannot form a triangle. Thus, the answer is that they do not form any triangle.